In this paper, we consider numerical approximation to periodic measure of a time periodic stochastic differential equations (SDEs) under weakly dissipative condition. For this we first study the existence of the periodic measure ${\rho }_t$ and the large time behaviour of $\mathcal{U}(t+s,s,x)≔\mathbb{E}\varphi \left(X_t^{s,x}\right)-\int \varphi d{\rho }_t,$ where $X_t^{s,x}$ is the solution of the SDEs and $\varphi$ is a test function being smooth and of polynomial growth at infinity. We prove $\mathcal{U}$ and all its spatial derivatives decay to 0 with exponential rate on time $t$ in the sense of average on initial time $s$. We also prove the existence and the geometric ergodicity of the periodic measure of the discretised semi-flow from the Euler–Maruyama scheme and moment estimate of any order when the time step is sufficiently small (uniform for all orders). We thereafter obtain that the weak error for the numerical scheme of infinite horizon is of the order 1 in terms of the time step. We prove that the choice of step size can be uniform for all test functions $\varphi$. Subsequently we are able to estimate the average periodic measure with ergodic numerical schemes.

在本文中，我们考虑在弱耗散条件下对时间周期随机微分方程 (SDE) 的周期度量的数值近似。为此，我们首先研究周期测度的存在${\rho }_{吨}$ 和大量的时间行为 $\mathcal{你}(吨+秒,秒,X)≔\mathbb{乙}\phi \left(X_{吨}^{秒,X}\right)-\int \phi d{\rho }_{吨},$ 在哪里 $X_{吨}^{秒,X}$ 是 SDE 的解和 $\phi$是一个平滑且多项式增长的测试函数。我们证明$\mathcal{你}$ 并且它的所有空间导数都以时间指数速率衰减到 0 $吨$ 在初始时间的平均意义上 $秒$. 我们还证明了 Euler-Maruyama 方案中离散半流的周期测度的存在性和几何遍历性，以及当时间步长足够小时（对所有阶都是一致的）任何阶的矩估计。此后我们得到无限视界数值方案的弱误差在时间步长上为 1 级。我们证明步长的选择对于所有测试函数可以是统一的$\phi$. 随后，我们能够使用遍历数值方案来估计平均周期度量。